Decentralized Admission Control and Resource Allocation for Power-Controlled Wireless Networks

Decentralized Admission Control and Resource Allocation for Power-Controlled Wireless Networks S lawomir Stańczak 1,2 joint work with Holger Boche 1,2, Marcin Wiczanowski 1 and Angela Feistel 2 1 Fraunhofer German-Sino Lab for () Berlin, Germany 2 Heinrich Hertz Chair for Faculty of EECS Technical University of Berlin 22 September 2009 1/43

Outline 1 Introduction 2 Physical-Layer Abstraction by Interference Functions 3 User-centric Approaches 4 Network-centric approaches Distributed Power Control Algorithms Incorporating QoS requirements 2/43

Wireless Networks Goal: Study resource allocation and interference management Focus: High data rates, low or moderate channel dynamics Energy supply is not a bottleneck. Wireless mesh networks, cellular networks 4/43

Wireless Channel Characteristics Radio propagation channel is unreliable. channel fading, path loss, channel conditions are time-varying... Power and bandwidth are limited. Wireless spectrum is a shared medium. Link capacities are elastic. Network cannot be regarded as a collection of point-to-point links. Performance is maximized by tolerating interference in a controlled way. Resource allocation and interference management are necessary. 5/43

Wireless Network Resources and Mechanisms Wireless resources: power, time, frequency, space, codes, routes... Mechanisms for resource allocation and interference management Multiple antenna techniques MAC: power control and scheduling routing... Cross-layer protocols 6/43

Applications Voice transmission Inelastic traffic: QoS requirements need to be satisfied permanently. Data applications (WWW browsing,e-mail,ftp) Low QoS levels are temporarily acceptable. Elastic traffic: Applications modify their data rates according to available resources in communication networks. 7/43

Quality of Service User-centric approaches (inelastic applications): Satisfy strict QoS requirements of applications permanently. Network-centric approaches (elastic applications): Maximize the aggregate utility as perceived by the network operator. Address the issue of fairness. QoS link 2 minimum total power max-min fairness max P k α kqos k (weighted sum optimization) Feasible QoS region max P k QoS k (best overall efficiency) QoS link 1 8/43

The focus of this talk Power control with some aspects of the physical-layer design. Applications Single-hop communication with K > 1 logical links (users) System Constraints Network Layer QoS Link Layer Physical Layer Power Control + Transmit/Receive Strategies Concurrent transmission (works with any scheduling protocol). Each user is decoded (single-user decoding). Combination with routing and network coding strategies possible. 9/43

Feasible QoS Region: F γ Given a channel, F γ is the set of all QoS values that can be achieved by means of power control with all links being active concurrently. QoS link 2 ω 2 F γ feasible point infeasible point ω 1 QoS link 1 Assumption: ω k QoS Downward-comprehensive Upper-bounded may be non-convex. depends on power constraints P. F γ depends on the physical-layer realization: Key properties of many multiuser systems are captured by interference functions. 11/43

Signal-to-Interference(+noise) Ratio (SIR) Strictly monotonic QoS-SIR map: γ : R R + For any ω F γ, there is a power vector p P such that γ(ω k ) = SIR k (p) = p k I k (p) transmit power interference function e.g. Gaussian capacity (in nats/channel use): γ(x) = e x 1, x 0. 12/43

Axiomatic Interference Function Standard Interference Functions (SIF), Yates 95 A1 Positivity: I k (p) > 0 for all p 0. A2 Scalability: I k (µp) < µi k (p) for any p 0 and for all µ > 1. A3 Monotonicity: I k (p (1) ) I k (p (2) ) if p (1) p (2). It models many practical interference scenarios. 13/43

Interference Function: Examples Linear interference function I k (p) = (Vp + z) k Matched-filter receiver SIC receiver Minimum interference function I k (p) = min uk U k (V(u)p + z(u)) k MMSE receiver 2 1 V 1,1 1 QoS 1 u (2) V 1,2 u (3) interference QoS 3 2 3 QoS 2 3 SIR user 2 u (1) G u (4) SIR user 1 14/43

Problem Statement Problem (QoS-based power control under SIFs) w > 0 p(ω) = arg min w T p p P(ω) P(ω) := { p R K + : k SIR k (p) γ(ω k ) }. QoS link 2 Minimum total power p 2 p(ω) is the minimum point of P(ω) Valid power set ω 2 Feasible QoS region p 1 = γ 1 I 1 (p) p 2 = γ 2 I 2 (p) ω 1 QoS link 1 p 1 Zander 92, Foschini 94, Yates 95, Ulukus 98, Bambos 00, Boche&Schubert... 16/43

Fixed-Point Iteration, Yates 95 If ω is feasible, then the concurrent iterations converge to p(ω), where γ k γ(ω k ). k p k (n + 1) = γ k I k (p(n)) Component-wise increasing (decreasing) if p(0) = 0 (p(0) P(ω)). Amenable to distributed implementation, scalable, works for any SIF But how should new users join the network without disrupting the connections of active users? 17/43

Admission Control Scheme User k is called active at time n if SIR k (p(n)) γ k. Define A n to be the set of all active users at time n and B n := K \ A n. Power control with active link protection (δ > 1) { δ γ k I k (p(n)) k A n (active users) p k (n + 1) = δ p k (n) = δ n+1 p k (0) k B n (inactive users) δ > 1 can be interpreted as protection margin. the larger δ, the faster power-up of the inactive users. δ cannot be too large for all users to be fully admissible. Bambos 00, Chee Wei Tan 09 (only I k (p) = (Vp + z) k ) 18/43

Properties of the admission control scheme Theorem Let I k be any standard interference function. Then, All SIRs converge to some values. All users are admitted in finite time if γ = (γ 1,..., γ K ) is feasible. Transmit powers are bounded if and only if δ γ is feasible. Active users (k A n ): Preservation of active users: A n A n+1. Bounded power overshoot: p k (n + 1) < δp k (n). Inactive users (k B n ): SIRs of inactive users are increasing SIR k (p(n)) < SIR k (p(n + 1)). Stanczak&Kaliszan&Bambos 09 19/43

No power constraints, TX and RX beamforming SIR 8 7 6 5 4 3 2 1 0 n 0 0 10 20 30 40 0 50 100 150 200 A.1 T.1 A.2 T.2 A.3 T.3 A.4 T.4 A.5 T.5 A.6 SIR 90 80 70 60 50 40 30 20 10 user active at n = 0 user inactive at n = 0 common SIR target A.x: Admission control phases T.x: Transceiver optimization phases n K = 10, n T = n R = 4, γ = 8, δγ = 9.6, A n = {1,..., 5} The highest feasible SIR (example): 0.88 (fixed beamformers), 1.37 (RX beamforming), 8 (TX/RX beamforming) 20/43

Incorporating power constraints Theorem Suppose that δγ is feasible and p(m) βδi ( p(m)) holds for some m N 0 and β [1, β max ]. Then, there exists β max > 1 such that A n A n+1 for all n m. Active users send distress signals until the condition is satisfied. Stanczak&Kaliszan&Bambos 09 21/43

Problem Statement Problem (Utility-based power control) ω = arg max w T ω w > 0. ω F γ ω is a maximal point of F γ ω 2 ω 2 w Maximal point of F γ Feasible QoS region F γ ω 1 F γ ω 1 Goodman 00, Saraydar 02, Xiao 03, Neely 03, Chiang 04, Huang 05,Tassiulas 05... 23/43

Convexity of Feasible QoS Region q2 w q = (q 1, q 2 ) F γ (P) q 1 Find a class of strictly increasing and concave utility functions Ψ with γ(ψ(x)) = x, x 0 so that F γ is a convex set. 24/43

Convexity of Feasible QoS Region: Sufficient Conditions Theorem (Convexity under a Linear Interference Function) If γ with γ(ψ(x)) = x, x 0, is log-convex, then the feasible QoS region is a convex set, regardless of the type of power constraints Observation:γ is log-convex if and only if Ψ e (x) := Ψ(e x ) is concave. Further related results: Log-convexity of γ(x) is necessary for the region to be convex in general. Convexity of γ(x) is sufficient if V is confined to belong to some subset of nonnegative matrices. 25/43

Examples of Function Classes Ψ α (x) = { x 1 α 1 α α > 1 log(x) α = 1 log x α = 1 log Ψ α (x) = x 1+x α = 2 log x 1+x + α 2 1 α > 2 j(1+x) j j=1 α = 1: Throughput maximization α : Max-min fairness 26/43

Arbitrarily Close Approximation of Max-Min Fairness Let ω k = Ψ α(sir k) and let ν k = log(1 + SIR k). Then, ν converges to the max-min rate allocation as α. Flow 1 Flow 2 Flow 3 Flow 4 2.4 Source Rates Sum of Source Rates 8.4 8.2 1.8 8 1 6 α 11 16 27/43

Efficiency of the Max-Min SIR Power Allocation Theorem If p and q are positive right and left eigenvectors of B (k0) = V + 1 P k0 ze T k 0, then (i) p is max-min fair power allocation if and only if p = p. (ii) ω is max-min fair ω if and only if w = w = q p > 0. ω 2 = log(sir 2) w max-min fairness w max ω Fγ(P) w T ω α 1 < α 2 < α 3 < α 4 α 4 α 3 α 2 α 1 = 1 ω 1 = log(sir 1) 28/43

Joint Power and Receiver Control Alternating optimization 1 Given U(t 1) compute p(t) 2 Given p(t) compute U(t) 29/43

Joint Power and Receiver Control Alternating optimization 1 Given U(t 1) compute p(t) (i) Compute the weight vector: w = y(b (m) ) x(b (m) ), m = arg max k ρ(b (k) ) (ii) Compute the QoS vector: ω = arg max ω Fγ w T ω (iii) p(t) = (I Γ(ω )V) 1 Γ(ω )z, Γ(ω) = diag(γ(ω 1),..., γ(ω K )) 2 Given p(t) compute U(t) (i) k u k (t) = arg max x 2 =1 SIR k (p(t), x) monotonic convergence to max-min SIR-balancing solution But: not amenable to distributed implementation Theory provides a basis for novel decentralized algorithms for finding a saddle point of the aggregate utility function. 29/43

Some Simulation Results 50 Static PC (Utility Max.) PC (Queue Weighted Utility Max.) 40 min k SIR k /γ k 40 35 30 25 20 15 10 5 Maximum power PC PC + RX Beamf. PC + RX/TX Beamf. average delay average delay 30 20 10 0 0 50 40 30 20 0.1 0.2 0.3 0.4 0.5 0.6 PC+Beamforming (Max Min SINR) PC (Utility Max.) PC (Queue Weighted Utility Max.) 0.7 0 10 20 30 40 50 number of users 10 0 0 0.2 0.4 0.6 0.8 arrival rate 1 1.2 1.4 30/43

Utility-Based Power Control Equivalent minimization problem: ψ(x) = Ψ(x) p = arg minf (p) = arg min w kψ ( SIR k (p) ). p P p P k Positivity of minimizers: p > 0 Even if ψ(e x ) is convex, the problem is not convex in general. 32/43

Convex Statement of the Problem Theorem If I k (e s ) is log-convex and ψ(e x ) convex, the following problem is convex: s := log p, p > 0 s = arg minf e (s) S := {log x : x P + } s S F e (s) = F (e s ) I k (e s ) = l v k,le s l + z k is log-convex (Hoelder inequality). 33/43

Gradient-Projection Algorithm Let τ > 0 be constant step size (small enough), and let ] s(n + 1) = Π S [s(n) τ F e (s(n)) s(0) S F e (s) = diag(e s1,..., e s K ) F (e s ): F (p) = (I V T Γ(p))g(p) g k (p) = w k ψ (SIR k (p))sir k (p)/p k (locally available) Γ(p) = diag(sir 1(p),..., SIR K (p)) 34/43

Min-max reformulation min s ( e s max w k ) kψ u k u k e s ˆp 0 subject to u t 0 k I k (e s ) t k = 0. Linear interference function: I k (e s ) = (Ve s + z) k. The Hessian is diagonal and its diagonals are given by the gradient. 35/43

Conditional Newton Algorithm 8 ><! s(n + 1) µ(n + 1) >: (u,λ u,λ,t)l(z(n + 1)) = 0! = s(n) ( 2 (s,µ)l(z(n))) 1 (s,µ) L(z(n)) µ(n) can be solved explicitely L(z) = L(s, u, µ, λ u, λ, t): A modified Lagrangian function. Quadratic convergence. Global convergence if ψ(x) = log(x) and ψ(x) = 1/x. No step size. Distributed implementation possible! 2 1 0 K = 50 ψ(x) = log(x) Conditional Newton Gradient projection -1 0 5 10 15 20 25 36/43

Adjoint Networks: A Simple Example Primal network: Measure SIR at E1 and E2 E1 and E2 compute some messages based on local measurements/parameters S1 S2 E1 E2 37/43

Adjoint Networks: A Simple Example Reversed network: The messages determine the transmit powers of E1 and E2. Cooperation by interference S1 and S2 estimate the messages from the received powers S1 S2 E1 E2 Significant gains compared to schemes relying on flooding protocols. Estimation errors are dealt with stochastic approximation. 37/43

Problem Statement s (ω) := arg min s S F e (s) s.t. k f k (s) := I k (e s )/e s k 1/γ k 0 ω2 F γ 00 11 000 111 000 111 0000 1111 00000 11111 00000 11111 00000 11111 000000 111111 000000 111111 000000 111111 000000 111111 P 2 p 2 000000 111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000 11111 P (ω) ω 1 P 1 p 1 The projection is not amenable to distributed implementation. 39/43

Primal-dual algorithm based on standard Lagrangian Primal-dual iteration under individual power constraints { [ s k (n + 1) = min s k (n) δe s ( ) ] } k (n) h k (s(n)) + Σ k s(n), µ(n), log(p k ) λ k (n + 1) = max { 0, λ k (n) + δf k (s(n)) } Σ k (s, λ) = l v l,k( λl e s l ) + SIR l (e s )g l (s) = l v l,km l (s, µ l ) Estimation of Σ k using the adjoint network. 40/43

Soft QoS Support F α (z) = ( SIRk (p) ) a k ψ α + b k ψ ( SIR k (p) ). γ k k A k B }{{}}{{} penalty term aggregate utility A: QoS users need to satisfy SIR k γ k, k A B: best-effort users A \ B: pure QoS users (voice) A B: best-effort users with QoS requirements (video) B \ A: pure best-effort users (data) Each user, say user k, determines its utility by choosing α k 1. Slightly modified algorithms Amenable to distributed implementation 41/43

Soft QoS Support: Example QoS of Pure QoS User 2 γ 2 1 - max-min-fairness 2 - Utility-based power control with α 2 = 1 3 - Utility-based power control with soft QoS support, α 2 1 A = 2 B = 1 No overshoot of user 2. 3 2 3 as α 2 QoS of Best Effort User 1 2 SIR 12 8 4 B \ A SIR Target A \ B A B a = b = 1 K = 4 SNR=40dB ψ(x) = log(x) B \ A 0 2 6 10 14 18 α 42/43

Foundations in Signal Processing, Communications and Networking Vol. 3 W. Utschick H. Boche R. Mathar Series Editors Fundamentals of Resource Allocation in Wireless Networks Theory and Algorithms Second Expanded Edition Sławomir Stańczak Marcin Wiczanowski Holger Boche This book series presents monographs about fundamental topics and trends in signal processing, communications and networking in the field of information technology. The main focus of the series is to contribute on mathematical foundations and methodologies for the understanding, modeling and optimization of technical systems driven by information technology. Besides classical topics of signal processing, communications and networking the scope of this series includes many topics which are comparably related to information technology, network theory, and control. All monographs will share a rigorous mathematical approach to the addressed topics and an information technology related context. The wireless industry is in the midst of a fundamental shift from providing voice-only services to offering customers an array of multimedia services, including a wide variety of audio, video and data communications capabilities. Future wireless networks will be integrated into every aspect of daily life, and thus could affect our life in a magnitude similar to that of the Internet and cellular phones. However, the emerging applications and directions require a fundamental understanding of how to design and control the wireless networks that lies beyond what the existing communication theory can provide. In fact, the complexity of the problems simply precludes the use of engineering common sense alone to identify good solutions, and so mathematics becomes the key avenue to cope with central technical problems in the design of wireless networks. That s why, two fields of mathematics play a central role in this book: Perron-Frobenius theory for non-negative matrices and optimization theory. Here these theories are applied and extended to provide tools for better understanding the fundamental tradeoffs and interdependencies in wireless networks, with the goal of designing resource allocation strategies that exploit these interdependencies to achieve significant performance gains. This revised and expanded second edition consists of four largely independent parts: the mathematical framework, principles of resource allocation in wireless networks, power control algorithms and appendices. The latter contain foundational aspects to make the book more understandable to readers who are not familiar with key concepts and results from linear algebra and convex analysis. SPCN_Stanczak.indd 1 isbn 978-3-540-79385-4 05.06.2008 17:12:41 Uhr Conclusions and Outlook The power control problem is relatively well-understood. Throughput maximization in the low SINR regime is an open problem. Outlook Joint optimization of transmit powers, schedulers (time+frequency), physical-layer (single- and multi-mode transmission). Dynamic optimization over finite and infinite time horizon. Stochastic power control.... springer.com Stańczak Wiczanowski Boche AB Fundamentals of Resource Allocation in Wireless Networks 2nd Ed. W. Utschick H. Boche R. Mathar Series Editors Foundations in Signal Processing, Communications and Networking Volume 3 Fundamentals of Resource Allocation in Wireless Networks Theory and Algorithms Second Expanded Edition Sławomir Stańczak Marcin Wiczanowski Holger Boche 1 23 43/43